In the scalar tensor theories of gravitation the Einsteinian cubic gravity was developed in Ref. [1]}, representing a particular extension which includes higher order contractions of the Riemann tensor. Such an approach has attracted attention in the last couple of years, being studied in various cosmological applications [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}. The extension towards a generic theory based on the cubic invariant has been proposed in [14]}, analyzed numerically. Later, such an approach has been studied [15]}, [16]} dynamically considering the linear stability theory, investigating the physical features of the phase space structure. The inclusion of the cubic component for scalar fields has appeared shortly [17]}, [18]} for tachyonic models and quintessence or phantom components. Furthermore, a more general extension which takes into account also the scalar curvature has been investigated [19]}. The observational analysis for the generic cubic gravity [20]} showed the regions of interest for various parameters from an astrophysical point of view. Moreover, various authors have investigated the inflation [21]}, [22]}, [23]}, [24]}, [25]} and the black hole solutions [26]}, [27]}, [28]}, [29]} in different cubic gravitational theories.
